As so many others have said, 1UPT. Civ 5 apologists like to pop up and point out that SoDs aren't a very good system either, but few here are arguing to the contrary. It's just that while SoD was a pretty crappy system, at least it worked within the context of the game.
1UPT is just awful. It just doesn't work as a Civ game mechanic, period. The problem with 1UPT is that your production capabilities are always going to vastly exceed the amount of soldiers you can actually field. This means that yes, you'll spend more time than you should with nothing useful to build, but it also means that an empire with powerful infrastructure won't have as significant of a military advantage as they should.
What you've gotta know is that we're not opposed to 1UPT on principle. Most of us here aren't just avid Civ players, but avid TBS players in general. And a lot of TBS games use 1UPT, either overall (Panzer General, Battle for Wesnoth) or just in battle (Age of Wonders, Heroes of Might and Magic, Master of Magic). It's a system we're familiar with and many of us are quite comfortable with.
No, the problem with Civ 5 is that the 1UPT system was implemented in a way that was completely incongruous with the rest of the game mechanics. I feel like the devs had no idea how it would impact the rest of the game, and in their haste to address one problem, they ended up causing many more.
There were a great many other issues with Civ 5 - lack of most features added in Civ IV (Civ IV built on Civ III's successes and experiments, why would Civ V not do the same?), awful AI, useless diplomacy, broken multiplayer - but many of these have been addressed since release. 1UPT is the one thing that constantly ruins the game and seems to have no fix in sight.
As for positives? Many say hexes, but to be honest, I felt like the difference was fairly superficial. To me, the one really cool thing Civ V did was Social Policies. Unfortunately, they are (or at least were) not very well balanced at all, and I would prefer to see them in addition to Civics/governments, rather than instead of. Nonetheless, they had some really interesting ideas in there, and it was a cool experiment.
City states are, IMHO, what great people and golden ages were to Civ III. An experiment that kind of wobbled and flopped around a bit and you're not really sure if you like it or not, but something that could, with some expanding and rebalancing, become a really cool gameplay feature (as GP and GA did in Civ IV). Maybe an expansion?
Civ IV has two issues in common with Civ V: UI and AI. Both games need some dramatic improvements in those two areas. The other big issue is, of course, SoD; while it works far better with the Civ mechanics than 1UPT, it is still far from ideal. Stack caps have been suggested and I believe attempted, but the hard part is programming AI that can deal with it.
Other than that, I would only add a few minor balance changes to certain civics, wonders and buildings (It'd be nice to see people actually run Serfdom or build Chicken Pizza, for example), and a slightly more streamlined and cost-effective espionage system, and there are mods that do those things.
I think that Civ V fundamentally has three issues: It did a lot of work trying to fix what wasn't broken in Civ IV (global happiness, social policies, etc), when it did try to fix what was broken they ended up making it worse (1UPT), and they ended up completely ignoring the elements of Civ IV that needed to be fixed the most (AI, UI), again, making things even worse. And I'm just left asking: Why would I play this game over Civ IV?